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A $C^2$-function $u$ is subharmonic if and only if the matrix (the complex Hessian) $$ \left( \frac{\partial^2 u}{\partial z_j \partial \bar z_k} \right)$$ has positive trace. It is plurisubharmonic if and only if the complex Hessian is positive semidefinite.
That is, f is harmonic on every complex line. A function f: U → R ∪ { − ∞} is plurisubharmonic, sometimes plush or psh for short, if it is upper-semicontinuous and for every a, b ∈ Cn, the function of one variable ξ ↦ f(a + bξ) is subharmonic (whenever a + bξ ∈ U ).
In mathematics, plurisubharmonic functions (sometimes abbreviated as psh, plsh, or plush functions) form an important class of functions used in complex analysis. On a Kähler manifold, plurisubharmonic functions form a subset of the subharmonic functions.
In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory. Intuitively, subharmonic functions are related to convex functions of one variable as follows.
6 de jun. de 2020 · The plurisubharmonic functions for $ n > 1 $ constitute a proper subclass of the class of subharmonic functions, while these two classes coincide for $ n= 1 $. The most important examples of plurisubharmonic functions are $ \mathop {\rm ln} | f ( z) | $, $ \mathop {\rm ln} ^ {+} | f ( z) | $, $ | f ( z) | ^ {p} $, $ p \geq 0 $, where ...
1 Introduction. Removable singularities of analytic objects, e.g., holomorphic or plurisubharmonic (psh) functions, or more generally, holomorphic maps and closed positive currents, are of classical interest and have been deeply studied. A good reference is Carleson [3], Siu [19], or Harvey and Polking [12].
Let us assume that subharmonic oscillations of frequency \(\omega / 3 \approx \omega_{0}\) have somehow appeared, and coexist with the forced oscillations of frequency \(3 \omega\): \[q(t) \approx A \cos \Psi+A_{\text {sub }} \cos \Psi_{\text {sub }}, \quad \text { where } \Psi \equiv \omega t-\varphi, \quad \Psi_{\text {sub }} \equiv \frac ...
